Best answer I could give you right now is to refer you to someone else- an Army Recruiter.
DO understand that Combat Arms, such as Infantry, Artillery, Armor, Engineer is a separate track from Combat Support- such as Ordnance, Supply, etc. In MOST cases (not all) a unit armorer is a relatively low level position. Armorers are trained in the care and maintenance of weapons- for instance, how do I fix my XM144E2 .177 caliber smoothbore spreical pellet projector (that's a BB gun, son- keep up with me now) when it fails to feed. If the problem is at a certain level of expertise (the ejector has a broken spring) and it can be fixed by replacing that spring, then you do that at the unit level. If the problem is that the flamistan has been framboozled, and the fix for that is to regulate the thingamajig, and that is beyond your level, the weapon gets sent up to the next level, or the next, or the next (maybe a depot level rebuild)
And despite all the jokes about recruiters- they DO have the most current info on schools, career tracks, etc. If a recruiter makes a promise IN WRITING- and the service fails to follow thru on that promise, the recruit is released from service.
Being a military armorer can be good training- it does NOT immediately translate into being a civilian gunsmith- who may do things like MAKE parts that are not avaialble, refinish wood, and work on a HUGE variety of arms. But it CAN be a good start.